Average Error: 6.8 → 2.2
Time: 2.1s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\frac{z}{t} \cdot \left(y - x\right) + x\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\frac{z}{t} \cdot \left(y - x\right) + x
double code(double x, double y, double z, double t) {
	return (x + (((y - x) * z) / t));
}
↓
double code(double x, double y, double z, double t) {
	return (((z / t) * (y - x)) + x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.8
Target2.3
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 6.8

    \[x + \frac{\left(y - x\right) \cdot z}{t}\]

  2. Simplified6.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
  3. Using strategy rm

  4. Applied clear-num6.8

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{t}{y - x}}}, z, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef6.8

    \[\leadsto \color{blue}{\frac{1}{\frac{t}{y - x}} \cdot z + x}\]

  7. Simplified6.2

    \[\leadsto \color{blue}{\frac{z}{\frac{t}{y - x}}} + x\]
  8. Using strategy rm

  9. Applied associate-/r/2.2

    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} + x\]

  10. Final simplification2.2

    \[\leadsto \frac{z}{t} \cdot \left(y - x\right) + x\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))